Topic 1: Indifference Curves

The topics in this Lesson present a bit more advanced material than was
built into the previous two Microeconomics Lessons. We begin with
indifference curve analysis. An indifference curve is presented in
Figure 1 below.

Suppose we measure an individual's consumption of commodity X and commodity
Y along the horizontal and vertical axes respectively and then arbitrarily pick
a point in the resulting (X , Y) space such as, for example, point A.
Now imagine that we label with a plus sign every point in the space that is
preferred to point A and then label with a minus sign every point in the space
that point A is preferred to. If we then draw a line that separates the plus
from the minus signs, we will obtain the indifference curve shown in the above
figure. The individual will be indifferent between all combinations of X and Y
indicated by the curve and will prefer all combinations above the indifference
curve to any combination on the curve. And any combination along the
indifference curve will be preferred to all combinations below it.

Since every (X , Y) combination will have an indifference curve
passing through it, we can add a third axis stretching upward from the bottom
left corner of the figure measuring the degree to which the individual's
preferences are satisfied, and visualize the infinitely many indifference
curves as representing a smooth surface that rises as the consumption of
commodities X and Y increase. We denote the degree to which preferences are
satisfied as the level of utility and assume that individuals
choose the combination of goods X and Y, among those available, that maximizes
their utility, with an increase in utility occurring whenever there is an
increase in the quantity of either X or Y consumed, holding consumption of the
other good constant.

Utility theory thus assumes that individuals have an internally consistent
set of preferences that do not change during the time-interval during which
we are analyzing their behaviour. In this sense we assume that individuals
are rational. Irrational behaviour is illustrated in Figure 2
below.

Suppose that an individual has indifference curves that cross, as in the case
of Curve #1 and Curve #2 above. This implies that the individual is
indifferent between combinations A and B and between combinations A and C.
As a result, he must be also indifferent between points B and C. But point B
has to be preferred to point C because it is above the indifference curve on
which point C is located. The individual is consuming more of both goods at
point B than at point C. The crossing of two indifference curves presents a
logical contradiction in the sense that the individual is behaving
inconsistently or, as we would say, irrationally.

Economists have often been criticized for their assumption that people
are rational. After all, we can think of many examples of people doing stupid
things. Irrational behaviour of friends and relatives and other people we
observe is part of the human condition. In this respect, however, it is
important to understand that economists' definition of rationality means
simply that individuals behave consistently, however stupid and irrational that
consistent behaviour might appear to others. And while it is clear that some peoples'
behaviour may be unstable through time, the economist has to assume that the
bulk of people whose behaviour is being analyzed have unchanged preferences
during the period over which the analysis is taking place.

In fact, without an assumption that people have consistent preferences that
do not change during the period being analyzed, no coherent analysis of social
behaviour would be possible---all that would be possible is a factual
delineation of what has happened in the past by historians who must carefully
avoid any interpretation of those observed facts. Of course, psychologists
and sociologists, and occasionally economists, will attempt to determine how
and why preferences change through time, but they too have to assume coherent
and internally consistent preferences that are capable of systematic
interpretation.

In the simple case portrayed in the two Figures above, economists assume that
an individual's utility can be expressed as a function of---that is, dependent
on---the quantities of commodities X and Y consumed. Mathematically, we can
write

1.
U = U(X , Y)

where U is the level of utility and the function
U(X , Y) states simply that the level of utility depends in some
fashion on the levels of commodities X and Y consumed by the individual. If
we want to get fancy and analyse a situation where the individual's preferences
change, we could expand the utility function by inserting between the brackets
an additional input, call it Z, that measures the forces causing preferences
to change, yielding the function U(X , Y , Z).
Analytical extensions of this sort are, of course, extremely difficult if not
impossible to successfully pursue.

The presentation of the utility function in Equation 1 is extremely
general---without additional specifications, the relationship denoted by
U(X , Y) could take any form. Several important features of the
utility function are always specified. First, as we noted above, increases in
the levels of X and Y always lead to increases in U . That is,
the partial derivatives of the utility function with respect to X and Y ---the
changes in U associated with in small changes in each of X and Y holding the
other constant---are positive. Mathematically, this imposes the two conditions

2.
∂U/∂X =
∂U(X , Y)/∂X > 0
and
∂U/∂Y =
∂U(X , Y)/∂Y > 0

where ∂U/∂X&nbsp is the partial derivative
of U(X , Y) with respect to X and
∂U/∂Y&nbsp is the partial derivative with respect to Y.
We refer to ∂U/∂X and ∂U/∂Y
as, respectively, the marginal utility of X and the marginal utility of
Y. Equations 2 specify that marginal utilities of X and Y are positive.

The second specified feature of the function U(X , Y) is
the principle of diminishing marginal utility. This says
that the marginal utility of X declines as the quantity of X increases
and the marginal utility of Y declines as the quantity of Y increases.
The slope of an indifference curve is the negative of the ratio of the
marginal utility of X over the marginal utility of Y. To see this, imagine
that the quantities of X and Y change by small amounts. The change in utility
specified in Equation 1 can then be expressed mathematically as

3.
dU = ∂U(X , Y)/∂X dX +
∂U(X , Y)/∂Y dY =
∂U/∂X dX + ∂U/∂Y dY

where the letter d preceding a variable denotes a small change
in that variable. Since the level of utility must be constant---that is
dU = 0 ---along an indifference curve, Equation 3
can be rearranged to yield

0 = ∂U/∂X dX + ∂U/∂Y dY

which can be further rearranged as

4.
dY/dX = − ∂U/∂X / ∂U/∂Y

where dY/dX is the slope of the indifference curve. The principle
of diminishing marginal utility implies that ∂U/∂X&nbsp, the
marginal utility of X, falls as the quantity of X consumed increases and that
∂U/∂Y , the marginal utility of Y, rises as the quantity of
Y consumed decreases. As can be seen from Equation 4, this implies that the
indifference curve gets flatter as the quantity of X consumed increases relative
to the quantity of Y consumed. Or, as we say, indifference curves are concave
outward, or convex with respect to the origin. The slope of the indifference
curve is called the marginal rate of substitution, which declines
as the quantity of X increases relative to the quantity of Y.

Of course, the amounts of commodities X and Y that the individual will be able
to consume depends on the level of that person's income. If the entire income
is spent on commodity X, the maximum quantity that can be consumed is given by
the distance between the origin and point B on the horizontal axis of Figure 3
below. If the entire income is spent on commodity Y, the maximum quantity that
can be consumed is given by the vertical distance between the origin and point A.
If the prices of the two commodities facing the individual are constant, the
ratio of the price of commodity X to the price of commodity Y is given by the
slope of the budget line running from point A to point B.

The optimal quantities consumed will be that combination of X and Y that puts
the individual on the highest possible indifference curve---that is, quantities
X0 and Y0 on the above Figure.
Note that the equilibrium quantities are those for which the slope of the
indifference curve equals the slope of the budget line---that is, where the
marginal rate of substitution equals the price ratio.

Now suppose that the level of the individual's income increases without any
change in prices. More of both commodities can now be consumed and the price
ratio does not change, so the budget line shifts outward with the new budget
line being parallel to the original one. The level of utility increases from
U0 to U1 and the individual's
consumption of the two goods increases to X1
and Y1 . At this point we must keep in mind that the
indifference map in Figure 3 assumes that both X and Y are normal goods---that
is, that indifference curve U1 is tangent to the new
higher budget line at a point to the right of output
level X0. In the case where X is an inferior good, this
tangency would be to the left of output level X0 and
the quantity demanded of commodity X would decline as a result of the increase
in income.

Finally, let us suppose that the price of commodity X falls, with no change in
money income. The results are shown in Figure 4 below.

If the individual were to spend his entire income on commodity X, the amount of
X purchased would now be higher. Since the price of good Y has not changed,
neither has the maximum possible consumption of that commodity. The fall in
the price of X has thus reduced the slope of the individual's budget line by
rotating it counter-clockwise around point A on the vertical axis. The new
utility maximum occurs at point c with a big increase in the quantity of
good X consumed and a slight decline in consumption of good Y.

It is important to distinguish between two components of the shift from the
initial equilibrium point to the final equilibrium at
point c---the income effect and the substitution
effect. The decline in the price of X leads to a substitution of good X
for good Y along the initial indifference curve, holding real income---that is
utility---constant. This substitution effect is indicated by the movement from
combination a to combination b along indifference
curve U0. The fact that real income has
increased as a result of the decline in the price of good X, holding nominal
income and the price of good Y constant, results in an increase in the
quantities consumed of both goods, represented by the movement from combination
b to combination c. This income effect is represented by the
movement from indifference curve U0 to
U1. As you can see from the above Figure, the quantity
consumed of good X increases as a result of both the substitution and income
effects while the quantity of good Y consumed declines as a result of the
substitution effect and increases by slightly less than that amount as a
result of the income effect, leaving a slight overall decline.

It should now be clear why demand curves slope downward when the goods, as in
the above analysis, are substitutes for each other. It is obvious from
Figure 4 that a fall in the price of commodity X, holding nominal income
constant, results in an increase in the demand for that good. In that Figure,
the fall in the price of good X also shifts the demand curve for good Y slightly
to the left because the substitution effect more than offsets
the effect of the decline in real income. Also, it is clear from Figure 3
that an increase in nominal income, holding prices constant, shifts the
demand curves of both goods to the right and, therefore, that both commodities
in that example are superior goods.

In the real world, each individual will spend her income on many
goods in each period of her life, and will face relative prices that may
change from period to period along with the interest rate, which measures the
cost of consuming in the present as opposed to future periods. In a world
where consumption externalities are present, she may also experience gains
and losses in utility from the behaviour of others over which she has no
control. This means that more advanced analysis will involve a utility function
with many more arguments. On the basis of our two-commodity analysis above,
however, it is reasonable to expect that the marginal rate of substitution of each
good for each other in the utility function will, in equilibrium, equal the
relative price of that pair of goods. The principles of diminishing marginal
utility and diminishing marginal rate of substitution can reasonably be
assumed to be widely applicable.

It is now time for a test. As always, think up your own answers before
looking at the ones provided.